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Violations of the Born rule in the Black Hole

Tim Faber, s1920014.

Supervisor: Dr. K. Papadodimas.

Second corrector: Prof. E.A. Bergshoeff

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In this master thesis we explore the possibility of violations of the Born rule postulate in the vicinity of the black hole. We introduce the black hole information paradox and review the recent development in the firewall discussion. In the second part we introduce a thought experiment in the framework of AdS/CFT. We formulate a conflict between quantum

entanglement and typicality. An exploration of this conflict by D. Marolf and J. Polchinski in 2015 [15] has resulted into the conclusion that ’non-excited’

black hole states could not be dual to typical CFT states without violating the Born rule. In this thesis we try to get more insight into the matter and explicitly quantify the statements made. By computing correlation function on the eternal BTZ black hole in 2+1 dimensions, we find violations of the Born rule to be immeasurable, indicating there would be a possibility for

non-excited black holes to be dual to typical CFT states.

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Contents

1 Introduction 1

2 Rindler Space 5

2.1 Two sets of coordinates . . . 6

2.2 The Boguliubov transformation . . . 9

2.2.1 Expansion of Rindler modes . . . 9

2.2.2 Solving the Boguliubov Coefficients . . . 12

2.3 Unruh Effect . . . 15

2.4 Hawking radiation and connection to black holes . . . 20

2.5 Information Paradox . . . 21

3 AdS/CFT 23 3.1 Introduction . . . 24

3.2 AdS . . . 27

3.3 CFT . . . 30

3.4 Relating the two sides . . . 32

3.5 Black holes in AdS/CFT . . . 33

3.5.1 Thermal Field theory . . . 33

3.5.2 Black Holes . . . 34

3.5.3 Two sided black hole in AdS/CFT . . . 37

3.6 Fields in AdS . . . 40

3.6.1 Fields in eternal BTZ . . . 43

3.7 Information problem in AdS/CFT . . . 47

3.7.1 Construction of the interior . . . 47

3.7.2 Quantum Cloning . . . 49

3.7.3 Black Hole Complementarity . . . 50

3.7.4 Strong Subadditivity and Firewalls . . . 51

3.8 Entanglement vs Typicality . . . 53

3.9 Quantum Chaos . . . 54

3.10 Violations of the Born Rule . . . 55

4 Violations of the Born Rule 57 4.1 The Born Rule . . . 57

4.2 Violations of the Born Rule for cool horizons . . . 57

4.3 Discrete modes vs Wave-packets . . . 60

5 Bounds of the Born Rule 62 5.0.1 A First Measure . . . 62

5.0.2 A Second Measure . . . 63

5.1 Relating measures . . . 64

6 Thought Experiment 65 6.1 A Thought Experiment . . . 65

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7.1.1 First order contribution . . . 74

7.1.2 Second order contribution . . . 76

7.2 Correlation Functions . . . 78

7.2.1 Two-point Function . . . 79

7.2.2 Pulse-approximation . . . 81

7.3 Horizon approximation . . . 83

7.4 Boundary approximation . . . 87

7.4.1 Change in Correlation . . . 90

8 Analysis 91 8.1 Horizon Approximation . . . 91

8.2 Boundary Approximation . . . 94

9 Discussion 96 10 Conclusion 98 11 Acknowledgement 99 12 Appendix A 100 12.1 Other operations? . . . 100

12.2 Schmidt Decomposition . . . 100

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1 Introduction

In 1972 Jacob Bekenstein connected the quantity of thermodynamic entropy to the area of black holes. The famous area law is given by the Beckenstein- Hawking formula[1]:

S = A

4GN (1)

This law relates the area of the black hole to the entropy. This statement made a very big impact. Elaborating on this calculation Stephen Hawking formulated his theory of black hole radiation and showed that black holes could evaporate [2]. The result caused a situation. A notorious situation known in the business as the ”black hole information paradox”. A black hole starting in a pure state would radiate away into a mixed state consisting of thermal radiation, causing to lose information of it’s purity. The principle of conservation of information or quantum unitarity was a sacred one, and never seen violated before. At the black hole, where quantum mechanics and general relativity met in concrete way, the situation looked very frightening. Either the process did not obey quantum unitarity, or something different happened which was at that moment, and for many years to come not understood.

A second interesting feature of Bekenstein-Hawking formula is the relation between the entropy and the area of the black hole. It suggested the idea of quantum holography. Originally formulated by Gerardus ’t Hooft [3]. The idea that the information inside a volume could be alternatively and fully described by the surface enclosing it. In 1998 J.C. Maldacena conjectured a concrete ex- ample of such a holographic principle. With string theory being the only true proposal as a theory for quantum gravity, he found a duality between a N = 4 supersymmetric Yang-Mills gauge theory without gravity, and a type IIB string theory compactified on AdS5× S5with gravity [4]. The boundary of the Anti- de Sitter space was found to be exactly equal to a quantum field theory with conformal symmetry, a conformal field theory or CFT. The AdS/CFT proposal was the first example of a gauge/gravity duality. An excellent tool to describe ill defined processes in one theory via the well understood dual theory.

The birth of quantum holography shed new light on the black hole infor- mation problem. Defining the process in AdS/CFT ruled out the possibility of information loss. However the solution to the paradox continued to be unknown.

Quite recently in 2009 S. Mathur, and later in 2013 by a group of researchers going by the name of AMPS (Almheiri, Marolf, Polchinski Sully), argued that the black hole information paradox redefined in AdS/CFT was a question of whether or not the interior region of the black hole existed at all[5,6]. In other words; If someone would try to cross the horizon, ’something’ should destroy that special someone. AMPS called this ’something’ a ’firewall’. Something quite dramatic seen in the light of general relativity, which according to the equivalence principle predicts the horizon to be a place like anywhere else.

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1 INTRODUCTION 2

The existence of the interior turned out to depend on whether or not the interior region of the black hole was entangled with the exterior region in a very specific way. This rather surprising, but very fundamental, fact gave birth to a formulation of a number of new paradoxes [7,8]. All originating from the newly explicification of the information problem in terms of quantum entanglement.

A black hole in AdS was shown to be dual to a very high energy state de- scribed by a thermal density matrix in the CFT [9,23]. Here one can connect the different micro-states of the ensemble in the CFT to the entropy of the black hole, all being entangled in different ways. In 2013 J. Polchinski and D. Marolf sharpened the paradox by stating that only very a-typical microstates, charac- tarized by a highly specific entanglement, would resemble black holes without a firewall, so called ’smooth’ horizon states [10]. They formulated a conflict between typicality and entanglement. Questions like, what kind of black hole then is the dual to a typical micro-state in CFT? Is there even one?

This conflict was honed even more by S. Shenker and D. Stanford. They brought in the phenomenon of quantum chaos [11,12]. With black holes known to be highly chaotic objects, the smallest perturbation was shown to have dramatic effects on the entanglement of the system. This highly specific entanglement which was needed for black holes to have smooth horizons turned from a very unusual unlikely case to a near impossible puzzle. Even so proposals for con- structing a black hole interior were developed [13,14], trying to go around and solving all the paradoxes that were lying out there. However in 2015 J. Polchin- ski and D. Marolf came back and stated that it in principle wasn’t possible to construct a AdS black hole with a smooth interior being dual to a typical CFT state within the laws of quantum mechanics. They posed that any construction of such kind would result into violations of the born rule postulate of quantum mechanics [15].

Now the Born rule has never shown to be violated anywhere in nature, nei- ther has it been been proven to exist. The Born rule however has been verified many times by experiment and is not considered as controversial. As being a postulate of quantum mechanics, it is a foundation on which the theory is build.

Taken to be true by assumption. If the born rule would be broken in the vicinity of a black hole, quantum mechanics would need a modification. Something very important to find out if searching for the real theory of quantum gravity.

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In this thesis we research this statement of Marolf and Polchinski; If one constructs an AdS black hole with a smooth horizon dual to a typical state in the CFT, do we observe violations of the Born rule. In section I, a broad introduction into the background theory is supplied. First, via a derivation of Rindler space, the semi-classical form of the information paradox is explained.

With this, the concept of Hawking radiation and the matter of entanglement between the inner and outer region of the black hole is analyzed. Continuing we briefly review AdS/CFT, the information problem in AdS/CFT, black holes in AdS/CFT, and the conflict between typicality and entanglement. We construct a black hole by applying quantum field theory on a black hole background in AdS/CFT. These results we can further use in our thought-experiment.

In the continuing section, II, we formulate a thought experiment to test the statement of Marolf and Polchinski. We define our black hole to have a smooth horizon and monitor the effect on the black hole state by acting on it with a unitary operator. This unitary operation will perturb the black hole state very mildly, however drastically change it’s entanglement configuration. We will try to interpret the modified state, and see what has happened both in the CFT and AdS picture. The question to answer now is, if the two pictures, AdS and CFT, show similar physical situations or do they differ. To that, if they differ, do they do so within the laws of quantum mechanics? We will derive a bound within quantum mechanics to test the born rule. We will present our results by computing correlation functions and the effect on the energy of the system. We conclude with a discussion, the implications of this research, and a look towards future research on the matter. The entire thesis uses c = ~ = G = k = 1

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1 INTRODUCTION 4

Section I

In section I is a theoretical background.. The review starts with the concept of Rindler space. Rindler space is flat space, however it functions as an excellent tool to understand what is going on at the black hole. As one will see very shortly, it shows many similarities with black holes, and can be used to grasp the concept of Hawking radiation and the entanglement issue related to black

holes.

Secondly the basics of AdS/CFT will be outlined. The AdS/CFT correspondence is the framework where computations will be made in. It is

therefore key to understand how one can describe the process of black hole evaporation. After this we can state the information paradox in it’s holographic form. The firewall arguments will be briefly summed, as the

connection without quantum chaos. When having done so, The conflict between entanglement and typicality can be best understood.

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2 Rindler Space

Rindler space is as mentioned above just ordinary Minkowski space. The key feature is presented by observing the spacetime symmetries of the metric. Ev- ery spacetime symmetry is generated by a ’Killing vector’, named after Wilhelm Killing. Minkowski spacetime has 10 different Killing isometries. 1 + 3 transla- tions, 3 rotations and 3 boosts. In ordinary (t, x) coordinates time translation is generated by the Killing vector ∂/∂tA Lorentz boost in the x− direction is sub- sequently generated by x∂/∂t+ t∂/∂x. However, if one changes coordinates to a boosted coordinate frame, time translation is now generated by x∂/∂t+ t∂/∂x. Both sets of coordinates will have a different notion of time, and therefore a different energy ground state or vacuum. This is called the ”Unruh effect” [21].

Relating both sets is done by a Bogoliubov transformation. This will be ex- plained below in detail, and gives rise to interesting phenomena in quantum field theory. A good full review on the subject is given by [16]. Below the same motivation for the Rindler construction is followed.

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2 RINDLER SPACE 6

2.1 Two sets of coordinates

The first set of coordinates will just be regular Minkowski coordinates. The Minkowski metric is given by:

ds2= dt2− dx2 (2)

With

x = (x, y, z) (3)

From now on we just consider x instead of x.

Consider now the following boost operation with boost parameter β:

t → t cosh β + x sinh β (4)

x → t sinh β + x cosh β (5)

One can observe that plugging the new coordinates back into the metric equa- tion, the metric stays the same. In other words the metric is invariant under this transformation. This isometry is generated by the boost Killing vector:

Kµ= x∂

t+ t ∂

x (6)

One could see that this symmetry suggests the following coordinate transforma- tion: Consider now the following transformation motivated by the boost Killing vector:

t = χ sinh η (7)

x = χ cosh η (8)

Resulting in the following diagram:

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Figure 1: Rindler space

As one can observe, the trajectory runs down in the quadrant II. This due to the fact that the boost operator works consequently on negative x. Secondly, the fact that the entire Rindler plane consists out of four wedges. The left, right, future and past wedge. For now we focus on the left and right wedge and later continue to the future and past regions. Both the left and the right Rindler wedge are can be related to the Minkowski plane.

t = eξa¯ sinh a¯η

a (9)

x = eξa¯ cosh a¯η

a . (10)

and for the left wedge:

t = eξa¯ sinh a¯η

a (11)

x = −eξa¯ cosh a¯η

a . (12)

only differing by a minus sign for the spatial component.These new coordinates will uniformly accelerate the old coordinates assymptoting to the speed of light.

The spacetime has become asymptotic and is now characterized by the two Rindler horizons on the lightcone at U = V = 0.

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2 RINDLER SPACE 8

The metric now looks like:

ds2= e2ξa(dη2− dξ2). (13) The metric is independent of η. The Killing vector describing time translation is now given by ∂/∂η. To describe our spacetime in quantum field theory we go first to lightcone coordinates: Minkowski coordinates can be related to lightcone coordinates in the usual way:

u = t − x (14)

v = t + x (15)

Also the Rindler coordinates (η, ξ) can be taken together in UV coordinates:

U = η − ξ (16)

and,

V = η + ξ (17)

We can relate the uv Minkowski coordinates to the UV Rindler ones as follows:

For the right wedge:

u = t − x = eξasinh aη

a −eξacosh aη

a = −1

ae−aU (18)

Where U = η − ξ. The same for v and V:

v = t + x = eξasinh aη

a +eξacosh aη

a = 1

ae−aV (19)

For the left wedge:

u = t − x = ea ¯ξsinh a¯η

a +eξa¯ cosh a¯η

a = 1

ae−a ¯U (20) Where U = η − ξ. The same for v & V:

v = t + x = eξa¯ sinh a¯η

a −eξa¯ cosh a¯η

a = −1

ae−a ¯V (21) We called the Rindler coordinates for the left wedge, ¯U and ¯V . In this case these coordinates are related to η and ξ like this:

U = ¯¯ η + ¯ξ (22)

and,

V = ¯¯ η − ¯ξ (23)

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2.2 The Boguliubov transformation

2.2.1 Expansion of Rindler modes

In the following section, we continue the story to add some quantum fields to the metric. We are going to do this according standard quantum field theory.

When doing so we have two notions of time. In the Minkowski coordinates time translation is generated by the usual Killing vector ∂/∂t, however in Rindler coordinates this is done by the above mentioned boost Killing vector x∂/∂t+ t∂/∂x. Now usually the positive frequency modes are defined to correspond to the annihilation operator in the mode expansion. These therefore define the notion of the vacuum state. However now that there are two notions of time- translation, two sets of annihilation operators with different sets of positive frequency wavefunctions, we have two different vacua. What happens below is the relation between these two sets of modes. This is done by a Boguliubov transformation and is possible since both are complete sets of energy-momentum eigenstates.

The expansion of the scalar field (eq. 9) can be written in two dimensions, 1 time and 1 spatial, as follows:

φ(x, t) = Z

−∞

√dk 4πk



a(~k)e−ıkµ(t−x)+ a(~k)eıkµ(t−x)



(24) or

φ(x, t) = Z

−∞

√dk 4πk



a(~k)fM(t, x) + a(~k)fM?(t, x)



(25) with kµ= (ω, ~k)

We can subdivide the integral in a negative and a positive part:

(26) φ(x, t) =

Z 0

√dk 4πk



a(k)e−ıkµ(t−x)+ a−†(k)eıkµ(t−x) + a+(k)e−ıkµ(t+x)+ a+†(k)eıkµ(t+x)



We can clean this up a bit by making use of the uv-coordinate change. Recall:

u = t − x and v = t + x φ(u, v) =

Z 0

√dk 4πk



a(k)e−ıkµu+ a−†(k)eıkµu+ a+(k)e−ıkµv+ a+†(k)eıkµv

 (27)

We can grab the u parts and v parts together and write the function like this:

φ(u, v) = φ(u) + φ+(v) (28)

We made a distinction between the positive and negative frequencies corre- sponding to left and right moving waves.

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2 RINDLER SPACE 10

The next step is to derive the same expression for fields expanded in Rindler coordinates. The KG equation in terms of Rindler coordinates (η, ξ), needs to be solved and these solutions have to be expanded in a similar way. Remember we work with a massless scalar field.

(∂tt+ ∂xx)φ = 0 (29)

now becomes:

(∂ηη+ ∂ξξ)φ = 0 (30)

Since the Lagrangian density of the field is invariant under this transformation the quantization procedure is exactly the same and we can express the field in terms of these new coordinates like this:

ψR(ξ, η) = Z

−∞

0

√ 4πω



bR(ω)e−ıωµ(η+ξ)+ bR†(ω)eıωµ(η−ξ)



(31) or

ψR(ξ, η) = Z

−∞

√dω 4πω



bR(ω)gR(η, ξ) + bR†(ω)gR?(η, ξ)



(32) Again similar commutation hold for these creation and annihilation operators of Rindler modes:

[b(ω), b(ω)] = iδ3(~x − ~y] (33) [b(ω), b0))] = 0 (34)

[b(ω), b(ω0)] = 0 (35)

Similarly we can write the expansion in in a positive and negative part. Making use of the coordinates U = η − ξ and V = η + ξ we end up with:

ψR(U, V ) = Z

0

√dω 4πω



bR(ω)e−ıωµU+ b−†R (k)eıωµU+ b+R(ω)e−ıωµV + b+†R (ω)eıωµV

 (36)

And in short splitted up into a positive and negative part for k:

ψR(U, V ) = ψ−R(U ) + ψR+(V ) (37) Now this expansion is only for fields in the right wedge, since these coordi- nates map to this wedge only. This is indicated by the R in the superscript of the creation and annihilation operators. For the left wedge the expansion looks the same only with coordinates (¯η, ¯ξ). The Rindler vacuum is defined as : bR(ω) |0Ri = bL|0Ri respectively for the right and left wedge.

Moreover due to this transformation symmetry in the Lagrangian density we can demand that the fields are equal to each other. We are going to work with the right wedge coordinates.

φ(u, v) = ξR(U, V ) (38)

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or

φ(u) + φ+(v) = ψ−R(U ) + ψR+(V ) (39) We consider only the positive parts, φ+(v) and ψ+(V ). We can split the left and right movers. These two can be taken separately, since they don’t interact with one another. Continuing:

φ+(v) = ψR+(V ) (40)

or:

Z 0

√dk 4πk



a+(k)e−ıkµv+ a+†(k)eıkµv



= Z

0

√dω 4πω



b+R(ω)e−ıωµV + b+†R (ω)eıωµV

 (41)

To solve this equation we take a Fourier transform to V on both sides. Or in other words: we multiply withR

−∞

dV0

e(ikV0). Let us first look at the right hand side of equation 48.

Right hand side:

(42) Z

−∞

√dV

2πe(iω0V ) Z

0

√dω 4πω



b+R(ω)e−ıωV + b+†R (ω)eıωV



= Z

0

√dω 2π

Z

−∞

dV0

√4πω



b+R(ω)e−ı(ω−ω0)V + b+†R (ω)eı(ω+ω0)V



By taking the Fourier transform to v we cancel to Fourier transform to k and end up with:

= 1

p2|ω|

 b+R(ω) for ω > 0

b+†R (ω) for ω < 0 (43)

Now the left hand side of eq. 48 is unfortunately less trivial. We start with the fourier transform to V.

Left hand side:

(44) Z

−∞

√dV

2πe(iω0V ) Z

0

√dk 4πk



a+(k)e−ıkµv+ a+†(k)eıkµv



= Z

0

√d 2π

Z

−∞

√dV 4πk



a+(k)eıω0V −ıkµv+ a+†(k)eıω0V +ıkµv



Because V is a Rindler coordinate and v a Minkowski one, we can try helping our cause by writing V in terms of v. Recall: v = 1ae−aV, and write:

Left hand side:

= Z

0

√dk 2k



a+(k)F (k, ω0) + a+†(k)F (−k, ω0)



(45)

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2 RINDLER SPACE 12

where,

F (k, ω0) = Z

−∞

dV 2πexp



0V + ik ae−aV



(46) We now set both sides equal to each other and find:

b+R(ω) = Z

0

dk



αRωKa+(k) + βωKR a+†(k)



(47)

The coefficients αRωk and βωKR are called the Bogulubov Coefficients, and are given by:

αRωk= rω0

kF (k, ω0) = rω0

k Z

−∞

dV 2πexp



0V + ik ae−aV



(48) and,

βωkR = rω0

kF (−k, ω0) = rω0

k Z

−∞

dV 2πexp



0V − ik ae−aV



(49) By solving these we find the relation between the Rindler operator b+r , and the Minkowski operators a+ and a+†.

We find a similar expression for b+R when we hermitian conjugate eq. 56:

b+†R (ω) = Z

0

dk



α?RωKa+†(k) + βωK?Ra+(k)



(50) Likewise we can find expressions for the Left Rindler wedge when substitute all R’s for L’s and use the transformation: v = −a1e−a ¯V.

Eq. 60 and Eq. 63 relate the operators between Minkowski and Rindler. It is also conveniant to have an expression that relate the modes. We can write down the following:

gR(η, ξ) = Z

0

dk



αRωKfM(t, x) + βωK?RfM? (t, x)



(51) and:

gR?(η, ξ) = Z

0

dk



α?RωKfM? (t, x) + βωKR fM(t, x)



(52) Another argument for this to be true is the completeness of both sets of modes.

2.2.2 Solving the Boguliubov Coefficients The next step is solving eq. (48), here we follow [17]:

αRωk= rω0

kF (k, ω0) = rω0

k Z

−∞

dV 2πexp



0V + ik ae−aV



(53)

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We can start by making a couple of substitutions. We take: x = e−aV, with dx = −ae−aVdV , s = −a and b = −ika

F (k, ω0) = 1 2πa

Z 0

dxxs−1e−bx (54)

Integration boundaries change to [0,∞), since x is a positive everywhere. αRωk now becomes:

αRωk = rω0

k 1 2πa

Z 0

dxxs−1e−bx (55)

The expression we have now, looks a lot like a gamma function. We use the identity: e−s log(b)Γ(s) = R

0 dxxt−1e−bx, where the logarithm is defined as:

log(A + iB) = log|A + iB|+i sgn(B) tan−1(|B|A ). Where0sgn0 is the sign func- tion, which gives the sign of a certain function. The sign function is 1 when the argument is positive, and negative when the argument is −1. So k > 0 gives a positive result etc. Now this is exactly what we need.

Proceding:

(56) αRωk =

0 k

1 2πa

Z 0

dxxs−1e−bx

= rω0

k 1

2πae−s log(b)Γ(s)

= rω0

k 1

2πaeiω0a log(−ika )Γ(−iω a)

Using the logarithm identity and some more algebra, we end up with:

(57) αRωk=

0 k

1 2πa

 a k

a

eωπ2aΓ(−iω0 a )

Here we have used in the exponent that the argument of the logarithm goes to

π

2, when the angle goes to infinity. It does so since the real part of the logarithm is zero.

Continuing by making use of another identity of the gamma function: xΓ(x) = Γ(1 + x), with x =−iωa 0 we get:

(58) αRωk =

0 k

1 2πa

 a

−iω0

 a k

iω0a eωπ2aΓ

 1 −iω

a



It is time to clean up. We end up with:

αRωk = ieωπ2a

√ kω2π

 a k

a

Γ

 1 −iω

a



(59)

Which is our final result for αRωk. The ω0 is changed for ω, since it was only a dummy variable.

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2 RINDLER SPACE 14

Computing the other Bogulubov coefficients goes on the exact same way: We find for:

βωkR = − ie−ωπ2a

√kω2π

 a k

a

Γ

 1 − iω

a



(60) Only the sign function changes sign since, we now have dealt with: F (−k, ω) instead of F (k, ω). We can find the following relation between the two coeffi- cients:

αRωk= −eπωa βRωk (61)

For the left wedge we deal we again take the positive ( ¯V ) part of the wavefunc- tion. By complex conjugating the right wedge we end up in the left wedge. For the rest we substitute L’s for the R’s and find:

αLωk = − ieωπ2a

√kω2π

 a k

a Γ

 1 + iω

a



(62) and,

βωkL = ie−ωπ2a

√ kω2π

 a k

a Γ

 1 +iω

a



(63) We can now relate these coefficients with each other and find the following relations between left and right.

βωkR = −eπωa α?Lωk, βωkL = −eπωa α?Rωk, (64)

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2.3 Unruh Effect

The first result we can derive is the following: The number operator is defined for Minkowski Operators by:

NkM = akak (65)

To give a trivial example on it’s significance, let us find the number of particles in the Minkowski vacuum or equivalently compute the expectation value of N in |0Mi

h0M| NM|0Mi = h0M| akak|0Mi = 0 (66) By definition, or eq. 13, we end up with 0. ak annihilates the ket-state, as does ak on the bra-state.

Now the same holds for the Rindler vacuum. Take the right wedge. We let the Rindler creation and annilihation operators work on the state and find:

h0R| NR|0Ri = h0R| bωbω|0Ri = 0 (67) The question is however what happens when we let the Rindler Operators work on the Minkowski vacuum. Right now with the Bogulubov coefficients we can make sense out of this question by relating bR and bR in terms of Minkowski operators and Bogulubov coefficients.

(68) h0M| NR|0Mi = h0M|

Z dk



α?RωKa(k) + βωK?RaR(k)



× Z

dk



αRωKa(k0) + βωKR a+†(k0)



|0Mi

= Z

dk|βωk|2h0M| aka0k|0Mi

= Z

dk|βωk|2δkk0

= Z

dk|βωk|2

We use our expression for βωk and end up with after some algebra:

h0M| NR|0Mi = 1

e2πωa − 1δ(ω − ω0) (69) This result is called the Unruh effect. It may seem surprising at first. Because what we see here is exactly the same expectation value as a Bose-Einstein parti- cle in a thermal bath of Temperature T = a/2π. This result might seem strange at first sight. If you consider the vacuum to be a state with zero energy then, yes indeed this result is strange. But this definition is not correct. The vacuum is the state with the lowest energy, and for Rindler space, as being accelerated spacetime, it is not so weird that the lowest energy state is different then the vacuum state of Minkowski space. Crucial to understand is the fact that only a Rindler observer is observing particles. Since this observer is accelerating.

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2 RINDLER SPACE 16

A Minkowski observer just freefalling through spacetime does not observe any particles, as the spacetime is still just flat Minkowski geometry.

To describe the path of a Minkowski observer in terms of Rindler modes one has to bring together the left and right wedge. This should be possible since both sets are complete and describe the entire spacetime. We make us the equation’s between the operators of the different expansions (44) and (47).

b+R(ω) = Z

0

dk



αRωKa+(k) + βωKR a+†(k)



(70) and,

b+†R (ω) = Z

0

dk



α?RωKa+†(k) + βωK?Ra+(k)



(71) Likewise for the left Rindler modes:

b+L(ω) = Z

0

dk



αLωKa+(k) + βωKL a+†(k)



(72) and,

b+†L (ω) = Z

0

dk



α?LωKa+†(k) + βωK?La+(k)



(73) We can now use the relations between the Bogoliubov coefficients, equation (61), to connect left and right ones. Substituting these in we obtain:

b+R(ω) = Z

0

dk



αRωKa+(k) − eπωa α?LωKa+†(k)



(74) ,

b+†R (ω) = Z

0

dk



α?RωKa+†(k) − αLωKa+(k)



(75) and for the left:

b+L(ω) = Z

0

dk



αLωKa+(k) − eπωa α?RωKa+†(k)



(76) ,

b+†L (ω) = Z

0

dk



α?LωKa+†(k) − αRωKa+(k)



(77)

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They are all written down explicitly since we will need a specific combination of them to let them describe the full Minkowski vacuum. We continue by looking at the expansion of the field in terms of Minkowski modes:

φ(v) = Z

0

√dk 4πk



a(k)fM(v) + a(k)fM? (v)



(78) We want to write a+(k) in terms of Rindler operators to see which combination of them annihilations the vacuum, or correspond to positive frequency modes.

In order to do so, we have to invert the previous equations and isolote all the parts that correspond to the a+(k) resp a+†(k). The way to find all operators being proportional to a+(k) is to take the following combination:

a+(k) ∝ b+R(ω) − eπωa b+†L (ω) This looks like this:

a+(k) = Z

0

dω C αRωK



b+R(ω) − eπωa b+†L (ω)



(79) where C is a constant given by: C = 1

1−e− 2πωa

. Furthermore α?L = −αR is used to relate the Bogoliubov coefficients. Of course, a similar relation is when L and R are interchanged:

a+(k) = Z

0

dω C αLωK



b+L(ω) − eπωa b+†R (ω)



(80) This relation can be written likewise for all negative frequency modes correspond to a+†(k) only switching to the hermitian conjugate of (76) and (77). Now we can derive a very important result from here. If we plug (76) and (77) into equation (75) we find that the following combination of Rindler operators should annihilate the Minkowski vacuum:

bR− eπωa bL†|0mi = 0 (81) bL− eπωa bR†|0mi = 0 (82) These relations imply the following:

bR†bR− bL†bL= |0Ri (83) This relation tells us that the number of Rindler particles in the left wedge is the same as in the right wedge. Right now we can write the following.

|0mi =Y

i

X

ni=0

Kn

ni!(bR†bL†)ni|0Ri (84) Here we follow [1]. We use a discrete sum instead of the integral to find the Kn. The physics don’t change by this. Continuing to find this parameter we use relations (eq. 85, 86) and find:

Kn+1− eπωia Kni = 0 (85)

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2 RINDLER SPACE 18

Solving:

Kni = eniπωia K0 (86)

Plugging back in:

|Omi =Y

i

X

ni=0

eniπωia K0

ni! (bR†bL†)ni|0Ri (87) We now define the state as follows: Every state with niparticles has niparticles with energy ωi in each wedge, left and right. Defining as in [1]

1

ni!(bR†bL†)ni|0Ri ≡ |ni, Ri ⊗ |ni, Li (88) and the state becomes per frequency(the product is left out):

|Omi = Ci

X

ni=0

eniπωia |ni, Ri ⊗ |ni, Li (89)

With normalization factor Ci= q

1 − exp(−2πωai). This is our main result from Rindler space. We see that the state of the Minkowski vacuum is an entangled state between the left and right Rindler wedge. This is a fundamental result and very important one for the continuing story. To underline the importance, when one wants to cross the Rindler horizon as a Minkowski observer the two wedges need to be in this exact entangled state. First, we can see what happens if one would put the system into a different state. For instance by perturbing the system and putting the system in a mixed state. The Minkowski vacuum now is described by the density matrix ρ. Lets see what we find: The density matrix of the system is now given by:

ρ = |ψi hψ| = |0Mi h0Mi| = (Ci)2

X

ni=0

e2niπωia



|ni, Ri ⊗ |ni, Li hni, R| ⊗ hni, L|

 (90)

We can write in terms of a left and right part by taking the partial trace:

ρR⊗ ρL (91)

The density matrix for the right wedge, ρR, is reached by taking the partial trace over the left eigenstates: in matrix form:

ρR=X

hni, L| ρ |ni, Li (92) and we end up with:

ρR= (Ci)2

X

ni=0

e2niπωia |ni, Ri hni, R| (93)

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What we see is a thermal state and correlation with the left wedge is lost. To make the situation even more concrete, we could see what happens with the stress energy tensor Tµν if the system is not in the entangled state (77). It is shown in appendix A that this quantity, Tµν, will be non-zero computed at the horizon. This means that there is energy sitting there! A first connection can be made with the concept of a ’firewall’.

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2 RINDLER SPACE 20

2.4 Hawking radiation and connection to black holes

The connection with Black holes is not very hard to see. In the case of Rindler space the acceleration has to be supplied by a rocket booster of some sorts, how- ever in the black hole case the gravitational field will do the trick. An immediate connection can be made describing an infalling observer in a Schwarzschild met- ric in terms of coordinate time and proper time. Probably known to the reader is the fact that the time needed to describe the trajectory in terms of coordinate time is infinite, while the elapsed time for an infaller to reach the horizon in terms of proper time is finite and well defined. The observer freely falling in terms of proper time is not accelerating away, however the one far away from the black hole trying to describe the process from outside of the black hole, has to accelerate away from the black hole to prevent him/herself from falling in.

Expanding the field in terms of modes for both observers gives the analogy to Rindler. Relating the asymptotic modes, often called outgoing modes, to the infalling modes in similar fashion as equation (69). We observe with now |ψi being the black hole vacuum for asymptotic modes:

hψ| bωbω|ψi = 1

eωT − 1δ(ω − ω0) (94) The Hawking flux or black body radiation spectrum for a black hole with tem- perature T , which is related to the acceleration by T = a [18]. By explicitly doing the calculation for a quantum field in d dimensions, one finds that this quantity is reduced by a so called grey-body factor. The fact is that modes can scatter of the gravitational field of the black hole. This causes a certain probability to exist for most to be reflected back towards to horizon, limiting the chance to escape completely.

hψ| bωbω|ψi = Γsωlm

e2πωa − 1δ(ω − ω0) (95) This gray-body factor can be seen as a transmission coefficient, and depends on the angular momentum of the mode [19]. The Hawking flux causes the black hole to evaporate since the energy of the Hawking photons is negative. The energy of these modes can be seen as the conserved charge corresponding to the time- translation Killing vector, which is time-like outside of the horizon. However this Killing vector becomes space-like inside the horizon. The conserved charge now becomes a momentum and can have a negative signature. It turns out, due to the mixing of positive and negative frequency modes in the interior, a negative sign is needed [18]. The description of the interior of the black hole is actually a highly relevant problem, which will be looked at in the AdS/CFT section, when describing the firewall argument.

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2.5 Information Paradox

With the mechanism of Hawking radiation explained, the original formulation of the information problem can be stated. Since the black hole loses it’s energy in the form of thermal radiation, the black hole will eventually end up in a complete mixed state of thermal radiation. However, the trouble arises if the black hole started out in a pure state. This process of black hole evaporation would cause a pure state to transit into a mixed state, and this cannot be described by a standard S-matrix process.

BHi → ρthermal (96)

To keep the process unitary, information has to travel outside of the horizon into the hawking radiation, which is forbidden by causality. As a consequence we end up with two possibilities. Either the information should be lost, or the Hawking radiation should in some way contain the information about the pu- rity of the state. This would mean that by computing all correlation functions between the Hawking photons, one would find the final state still to be pure.

As is the case for the process of burning up a pure state encyclopedia.

The situation can be viewed at graphically by looking at the von Neumann entropy of the system. 1 Now describing the two scenarios once more. What Hawking proposed was a linear increase in entanglement entropy. The black hole starting out in a pure state will slowly increase it’s entropy by the evap- oration process and do so until there was nothing but thermal radiation left.

The information preserving alternative needs the entanglement entropy to go to zero at the end of it’s lifetime. In other words, there has to be some tipping point where the entropy would start decreasing. This moment in time is called the Page time [41]. We sketch both cases below.

1The von Neumann or entanglement entropy is a measure to quantify the entanglement and is given by: S = − Tr ρ log ρ. A pure state will have zero entropy, while a mixed state will have maximal von Neumann entropy.

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2 RINDLER SPACE 22

Figure 2: The von Neumann entropy versus time for an evaporating black hole according to Hawking and Page3

A unitary black hole evaporation process follows the Page curve. Questions like, what triggers the entropy to decrease, or how does the full S-matrix of the black hole look like, are unanswerable at the moment. A full description of quantum gravity has to give insight in these puzzles, which is a wish for many.

For completeness of the review, we mention a third option for the evapora- tion process. The black hole could decrease to a remnant. A Planckian size object with a very high entropy. The entanglement entropy of the object would be so large, it would exceed the Bekenstein entropy and therefore violate the fact that the number of microstates is given by the Bekenstein entropy [19].

This last option seems very implausible, however was considered by some and has to be mentioned when discussing the information paradox.

The paradox seemed/seems very solid, and indeed nobody could crack the code for over twenty years. However new hope glared on the horizon, when the AdS/CFT correspondence came into our world.

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3 AdS/CFT

The black hole information paradox has gotten new light since the AdS/CFT duality was developed in 1998. In this review the basics of AdS, CFT and the connection between them is explained. After this we go back to black holes, talking about how one can desribe them in AdS/CFT. By applying quantum field theory to a BTZ-AdS metric, we will be able to construct a set up, on which we can compute correlation functions, and perform calculations on the black hole.

Secondly, the black hole information paradox in AdS/CFT is reviewed. We make the connection between typicality and entanglement. Furthermore the connection with quantum chaos is underlined, as it is an important factor in the conflict.

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3 ADS/CFT 24

3.1 Introduction

A hologram is a very thin film, describing a three dimensional object. The two dimensional film containing all the information of the object, only in an alternative description. The original idea of ’t Hooft [3] to suggest this might be relevant in physics was based on the work of Bekenstein and Hawking [1]

[2]. Every bit of information fallen into the black hole was described on the surface of the object. With entropy usually being thought of as a quantity related to the volume, this was an extraordinary idea. ’t Hooft proposed now to look at any closed surface area. He showed that the degrees of freedom inside were maximized if the area consisted of one big black hole. In the search for a theory of quantum gravity, black holes were considered to be a natural physical cut-off for quantum field theory to break down at higher energies. According to ’t Hooft it was therefore logical to describe any volume less energetic then a black hole, like ordinary quantum field theories, with a description lying on the surface enclosing that volume.

Susskind then proposed the holographic principle could be realized inside string theory [22]. Five years later the development of AdS/CFT in 1998 was the first realization of such a holographic principle. The original statement of the duality is given by[4]:

D = 4 ,N = 4 U(N) Super Yang Mills = IIB string theory on AdS5× S5 (97) where N is the rank of the field theory andN is the number of supersymme- tries. A supersymmetric Yang Mills theory in four dimensions is found dual to a string theory on a five dimensional Anti-de Sitter space times a five dimen- sional sphere. The gauge theory is subject to a conformal symmetry. Together with Poincar´e symmetry it is invariant under scale-transformations/dilations and special-conformal transformations. This high degree of symmetry field the- ory was found to be equivalent to the boundary of AdS space, which is highly symmetric manifold itself. It is characterized by a negative curvature.

It is illuminating to look at the couplings on both sides, following the review of [21]. The gauge theory is charactarized by the coupling, given by gY M and the rank of the fields, N . Again ’t Hooft showed that in the limit when N is large, one can perturbatively expand in terms of 1/N and g2Y M. The amplitude now has the following form:

Z =X

g≥0

N2−2gX

n=0

Cg,nλn (98)

where λ = g2Y M the ’t Hooft coupling.

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This can be identified with the loop expansion from string theory. The loop expansion in Riemann surfaces for a closed string theory has a similar form as equation (98) [9].

Z =X

g≥0

g2g−2s Zg (99)

with string coupling gs.

If we now want to relate both sides, we have to relate the parameters describing them. Next to the string coupling gs, which indicates the importance of quantum corrections, the string theory is defined by the curvature length LAdS and the string length ls. The ratio of the two, LAdS/ls, is a measure of how big the radius of AdS is in string lengths. If your curvature length is comparable to your string length, stringy/planckian effects are important.

For the duality to hold we find the following relations between the two sides:

gs= gY M2 ∼ λ

N,  LAdS ls

4

= 4πgY M2 N ∼ λ (100) We can observe the fact that the parameters can be tuned in the that is desir- able. Let us find the regime for classical gravity. For the stringy effects to be negligible, we need LAdS/ls>> 1. This means that we find λ >> 1. Secondly, we want quantum corrections to be small, therefore we want gs to be small, which means we need to take next to λ also N >> 1. We observe a very im- portant property of the duality. When evaluating the gravity side in the weak coupling regime, the gauge side turns out to be in the strong coupling regime.

Similar in the opposite case. The AdS/CFT correspondence is what is called a weak/strong coupling duality. One of the two sides is evaluated in the weak coupling regime, when the other is evaluated in the strong regime. This is what makes the duality so valuable. When on one side perturbation theory breaks down, one can observe what is happening on the other side.

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3 ADS/CFT 26

There is a lot of evidence to be found for the correspondence [27]. First of all, the symmetry groups on both sides agree. The isometry group of AdS5 is SO(4, 2), which matches with the conformal group in 4 dimensions4 After the discovery, Edward Witten and others elaborated on Maldacena’s work and started constructing a map between bulk fields and boundary operators [23,24].

This was the beginning of the so called ”dictionary”. A vademecum to relate quantities in the bulk to the boundary. The dictionary will be reviewed in sec- tion 3 of the AdS/CFT paragraph. With this, things like correlation functions [23], and causality [25,26], were tested, always with success. However, quanti- ties like correlation functions can be hard to compute in the strongly coupled regime of the CFT. This causes a limitation on what is testable. To emphasize once more, not once the correspondence seemed to be violated. The ground on which the duality seemed to stand looks pretty solid.

After the first example from Maldacena, many different dualities were developed in all kinds of dimensions. In 2009 the general formulation of the duality was stated by Polchinski et al. [31]. The volume/bulk theory is always described with one extra dimension, with on the boundary lying the conformal field the- ory. The general correspondence is usually formulated as AdSd+1/CF Td. The S is a trivial part of the duality, and is therefore left out of the formulation.

Figure 3: The conformal field theory lies on the boundary of the AdS cylinder.

Different cross sections in AdS correspond to different circles on the CFT

4For p > 1, q > 1, CO(Rp,q) ∼= SO(p + 1, q + 1), which for p = 1, q = 3 is equal to SO(4, 2) [27]

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3.2 AdS

The bulk is described by an Anti-de Sitter space. AdS is different from Minkowski space. It is a solution of the Einstein equation with a negative cosmological con- stant. The space is contracting. It is maximally symmetric. The isometry group of AdS5has 15 elements. AdSd is a d dimensional hyperbolic manifold, and can be described by a hyperboloid in d + 1 dimensional flat space. A set of points (X1, X2, ..., Xd+1) obeys the following equation:

−(X1)2− (X2)2+ ... + (Xd−1)2+ (Xd)2+ (Xd+1)2= −L2AdS (101) with L again the curvature length/radius of AdS. These points are embedded in an d + 1 dimensional space with metric:

ds2= −(dX1)2− (dX2)2+ ... + (dXd−1)2+ (dXd)2+ (dXd+1)2 (102) We can do several coordinate transformations to get more grip on how the space looks like. Often AdS is described by the Poincar´e patch. We follow the following coordinate transformations:

X1= 1

2z(z2+ L2AdS+

d

X

i=3

(xi)2− t2) (103)

X2= LAdSt

z (104)

Xi= LAdSxi

z (105)

Xd+1= 1

2z(z2+ L2AdS

d

X

i=3

(xi)2− t2) (106)

The Poincar´e patch is given by:

ds2=L2AdS z2



− dt2+ d¯x2+ dz2



(107)

with ¯x = Pd

3xi. The patch only describes part of the entire spacetime, since it is now singular at z = 0, therefore it only describes values for z 6= 0. The patch consists of Minkowski space slices ”warped” in the z-direction. After a conformal rescaling we can find for z → 0:

ds2CF T = −dt2+ d¯x2 (108) which is the expected Minkowski metric if considered in 4 dimensions.

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3 ADS/CFT 28

Another very useful coordinate transformation is the global patch. We have to transform equation (102) with the following coordinate changes. We switch to spherical coordinates for the d − 2 sphere. We introduce angles α3 until αd

z = L2AdS

pL2AdS+ r2 (109)

x3= 1

L2AdScos α1 (110)

xi= 1

L2AdSsin α3... sin αi−2sin αi−1 (111) xi−1= 1

L2AdSsin α3... sin αi−2cos αi−1 (112) We apply this to the Poincar´e patch, and obtain AdS in d in global coordi- nates:

ds2= −(1 + r2

L2AdS)dt2+ 1 1 + Lr22

AdS

dr2+ r2dΩ2d−2 (113) The time coordinate t is now the proper time in the center of the cylinder at r = 0 and runs from (−∞, ∞). The radial coordinate is zero at the center of the cylinder and runs to infinity at the boundary. The radius of AdS is related to the coupling λ.

Global coordinates are ’static’ coordinates. This means the spacetime now stays in one place. However, since AdS has a negative curvature, there is a potential towards the center. Because of this reason, AdS is sometimes thought of as

’gravity in a box’.

As one can observe the metric is of similar form of for example the Schwarzschild metric:

ds2= −f (r)dt2+ f (r)−1dr2+ r2dΩ2d−1 (114) Often another transformation to tortoise coordinates is made. Taking ρ = tan r running from [0, 2π], we write the metric in the following form:

ds2= 1 cos2ρ



− dt2+ dρ2



+ sin2ρdΩ2d−1 (115) Where we have taken lAdS = 1. This metric is convenient when making calcu- lations on the AdS metric, and will be used later on when expanding quantum fields on it.

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